Funny- we always used 6x6 or steel posts wrapped with 1x. 4x4, unless
every
4 feet with angle braces or something, is just too flexible for anything
more than a couple feet tall.

I just got back from looking at a cottage where a room was built over a
patio, supported by 4 4x4s. I didn't stay under it any longer than I had
to, and certainly didn't go in the room.
They would probably argue that it was cantilevered out, and the joists were
the real structural element, but why not use 6x6s?